A method for cleaning and/or disinfecting a milk-carrying system of a coffee machine for producing coffee mix drinks, and a coffee machine for carrying out the method has the aim of allowing the entire milk-carrying system to be cleaned, completely and in a way that can he easily automated. The milk delivery line is separated from the milk container, for carrying out the cleaning Process, and is connected to a first container, and a second container is arranged under a milk outlet of a milk preparation device of the milk-carrying system. Fluid is then delivered to fluid inlet in order to convey the active cleaning and/or disinfecting substance through the milk delivery line and a milk outlet into the second container. Thereafter, water is conveyed to the air inlet and the fluid inlet is closed, in order thereby to flush the milk-carrying system.
